Number Bases (Radices)

In mathematical numeral systems, the radix or base is the number of unique digits, including zero, used to represent numbers in a positional numeral system. For example, for the decimal system (the most common system in use today) the radix is ten, because it uses the ten digits from 0 through 9. Source

Today I turned 24793 days old, a prime number of days, and as it turns out a palindrome in base 12 (12421). I started playing around with the representation of 24793 in other number bases or radices. In base 31, the representation is poo so I may be in for a shitty day. In base 36, the letters of the alphabet are exhausted:

Beyond base 36, the following system is used:

More generally, in a system with radix b (b > 1), a string of digits $d_1\dots d_n$ denotes the number $d_1{b}^{n-1}+d_2{b}^{n-2}+\dots +d_n{b}^0$, where $0\le d_i<b$.

In practice, a colon is used to separate the individual digits e.g. 24793 is written$18:4:3{}_{37}$  in base 37 which can be dispensed with of course in the case of base 10.

Radices are usually natural numbers but they don't have to be. For example, a base using the golden ratio$\mathrm{\Phi }$ is possible, remembering that$\mathrm{\Phi }$ is given by the expression: $$\frac{1+\sqrt{5}}{2}$$ Commonly, the capital letter$\mathrm{\Phi }$ is used to represent 1.618033988749895… and the lower case letter $\varphi$ to represent 0.618033988749895… or$\mathrm{\Phi }-1$ but this is certainly not always the case.

Such a base is colloquially called phinary and the following table gives some idea of how it works (source) but uses$\phi$, a variation of the lowercase$\varphi$ but meant to equal 1.618033988749895… here:

$\mathrm{\Phi }$ is closely linked to the Fibonacci sequence since $$\underset{n\to \mathrm{\infty }}{lim}\frac{F_n}{F_{n-1}}=\mathrm{\Phi }$$More information about $\mathrm{\Phi }$ as a number base can be found on this site.